
where \(f_\theta \)  is a neural network with parameters \(\theta \)  and where \(\varepsilon _y\sim \mathcal {N}(0, \sigma _y^2)\)  is some normally distributed noiseThe index \(y\)  should not be misunderstood as \(\varepsilon _y\)  or \(\sigma _y\)  being \(y\)  dependent, but is supposed to indicate that the noise corresponds to the output. The same is true for \(\varepsilon _x\)  and \(\sigma _x\) , which we introduce in (REF ) below. that disturbs the true label (i.e., value of the regression function) \(f_\theta (x)\)  corresponding to \(x\) . Learning the neural network means to infer a value for \(\theta \)  from pairs \((x,y)\)  contained in a training set \(\mathcal {D}\) . The trained neural network can then be used to make predictions.
In real-world applications, especially in those where reliability and safety are crucial [1]}, [2]}, [3]}, it is valuable, if not indispensible, to know the uncertainty behind a prediction.
